A numerical study of interface dynamics in fluid materials

Abstract

This paper deals with the approximation of the dynamics of two fluids having non-matching densities and viscosities. The modeling involves the coupling of the Allen-Cahn equation with the time-dependent Navier-Stokes equations. The Allen-Cahn equation describes the evolution of a scalar order parameter that assumes two distinct values in different spatial regions. Conversely, the Navier-Stokes equations govern the movement of a fluid subjected to various forces like pressure, gravity, and viscosity. When the Allen-Cahn equation is coupled with the Navier-Stokes equations, it is typically done through a surface tension term. The surface tension term accounts for the energy required to create an interface between the two phases, and it is proportional to the curvature of the interface. The Navier-Stokes equations are modified to include this term, which leads to the formation of a dynamic interface between the two phases. The resulting system of equations is known as the two-phase Navier-Stokes/Allen-Cahn equations. In this paper, the authors propose a mathematical model that combines the Allen-Cahn model and the Navier-Stokes equations to simulate multiple fluid flows. The Allen-Cahn model is utilized to represent the diffuse interface between different fluids, while the Navier-Stokes equations are employed to describe the fluid dynamics. The Allen-Cahn-Navier-Stokes model has been employed to simulate the generation of bubbles in a liquid subjected to an acoustic field. The model successfully predicted the size of the bubbles and the frequency at which they formed. The numerical outcomes were validated against experimental data, and a favorable agreement was observed.